Method for calibrating a multi-sensor system using an artificial neural network

ABSTRACT

A method for calibrating a sensor with respect to a reference trajectory indicating a number of timely successive data values using an encoder/decoder neural network with an encoder function and a successive decoder function. The method includes training the encoder function and the decoder function of the encoder/decoder neural network with respect to a reference trajectory so that a reconstruction error of the reference trajectory at an input of the encoder function and a reconstructed reference trajectory at an output of the decoder function is minimized; training a calibration vector with respect to a sensor trajectory so that a reconstruction error between an output sensor trajectory and a corresponding reference trajectory is minimized, the output sensor trajectory is obtained by applying the calibration vector on the sensor trajectory and the encoder function and decoder function on the the calibrated sensor trajectory; and applying the calibration vector to calibrate the sensor.

CROSS REFERENCE

The present application claims the benefit under 35 U.S.C. § 119 ofEuropean Application No. EP 19176913.2 filed on May 28, 2019, which isexpressly incorporated herein by reference in its entirety.

FIELD

The present invention relates to multi-sensor systems with a pluralityof sensors for sensing correlated measures. The present inventionparticularly relates to methods for calibrating sensors in amulti-sensor system.

BACKGROUND INFORMATION

For many tasks, such as localization, mapping, tracking, and the like,multi-sensor systems are applied. In a technical system, multiplesensors are often used which are arranged offset from each other toprovide at least partly redundant and/or correlated sensor readings. Forinstance, multiple position and inertia sensors may be used on differentpositions on a robot which provide correlated sensor readings when therobot moves.

The quality of carrying out a specific task using the sensor readings ofthese multiple sensors is extremely sensitive to the calibration of thesensors used for the task. For sensors measuring the same or relatedmeasures, particularly, sensor orientation and position as well as theiraccuracy strongly affect the measurement results and essentiallydetermine the performance of the technical system. If the exact relationbetween the multiple sensors, such as their relative orientation andposition, is initially unknown, a sensor calibration is required toimprove the functionality of the technical system.

Conventionally, one approach to calibrate sensors of a technical systemincludes manually adapting calibration parameters of the sensors. Manualcalibration has the known disadvantages of inaccuracy and long duration.

A method for automatic calibration is, e.g., described in N. Schneideret al., “RegNet: Multimodal Sensor Registration Using Deep NeuralNetworks”, https://arxiv.org/pdf/1707.03167.pdf, wherein by means of afirst deep convolutional neural network (CNN) a 6 degrees of freedom(DOF) extrinsic calibration between multimodal sensors is carried outwherein the conventional calibration steps (feature extraction, featurematching and global regression) are performed in a single real-timecapable CNN.

Furthermore, G. Iyer et al., “CalibNet: Self-Supervised ExtrinsicCalibration using 3D Spatial Transformer Networks”,https://arxiv.org/pdf/1803.08181.pdf, describes a self-supervised deepnetwork capable of automatically estimating the 6-DoF rigid bodytransformation between a 3D LiDAR and a 2D camera. The network istrained to predict calibration parameters that maximize the geometricand photometric consistency of the input images and point clouds.

SUMMARY

According to the present invention, a method for calibrating one or moresensors of a multi-sensor system, a calibration system and amulti-sensor system are provided.

Further embodiments and developments of the present invention aredescribed herein.

According to a first aspect of the present invention, an example methodfor calibrating a sensor with respect to a reference trajectoryindicating a number of timely successive data values using anencoder/decoder neural network with an encoder function and a successivedecoder function is provided, comprising the steps of:

-   -   Training in a first training phase the encoder function and the        decoder function of the encoder/decoder neural network with        respect to at least one reference trajectory so that a        reconstruction error of the reference trajectory at an input of        the encoder function and a reconstructed reference trajectory at        an output of the decoder function is minimized,    -   Training in a second training phase a calibration vector, and        particularly the encoder function and the decoder function of        the encoder/decoder neural network, with respect to a sensor        trajectory so that a reconstruction error between an output        sensor trajectory and a corresponding reference trajectory is        minimized, wherein the output sensor trajectory is obtained by        applying the calibration vector on the sensor trajectory and by        applying the encoder and decoder function on the calibrated        sensor trajectory; and    -   Applying the calibration vector to calibrate the sensor.

In a sensor system with multiple sensors with correlated sensorreadings, such as a plurality of distributed sensors for sensing aposition or a movement of a movable system, sensor offsets, noise,sensor inaccuracy, external interventions and the like make theconventional approaches to calibrate the multiple sensors of themulti-sensor system complex or even impossible for high precisioncalibration.

Generally, for calibration, sensor trajectories of the sensors to becalibrated are used and brought into relation towards a referencetrajectory indicating the ground true system behavior during the sametime frame. General issues for calibrating sensors using sensortrajectories of sensor readings are the unknown local relations, such aspositional offsets of the sensors and the different time domains or thesensor readings, i.e., that each single sensor reading might have acertain off-time delay so that the sensor readings from differentsensors might not match exactly on a specific time step.

Matching trajectories with different off-time delays in combination withsensor noise, unknown local sensor distribution, and sensor inaccuraciesare common constraints when a multi-sensor system shall be calibrated.

The above example method provides an approach using an artificial neuralnetwork which can handle the complex environmental dynamics ofuncertainties and noises of a multi-sensor system.

Assuming the calibration of each sensor can be expressed by acalibration vector, which is configured with one or more dimensions. Thecalibration vector can be applied on a sensor trajectory to transformthe sensor reading, i.e., sensor trajectory of one of the calibratedsensors, to the reference trajectory, such as a sensor trajectory of areference sensor of the multi-sensor system obtained within the sametime frame, it is a task to automatically calibrate the sensors of thesensor system by adapting the respective calibration vector. So, eachsensor trajectory of each of the sensors to be calibrated can be appliedwith the respective calibration vector and thus be brought in line withthe reference trajectory of e.g. the reference sensor by means of thecalibration vector.

The above method applies a recurrent neural network as an LSTM(Long-Short Term Memory) encoder/decoder neural network. The LSTMencoder/decoder neural network includes an encoder LSTM unit to compresscalibrated trajectories of sensor readings of a specific sensor to becalibrated and to obtain an intermediate vector representing a lowerdimensional manifold. The calibrated trajectories are a result ofcalibration of the sensor trajectories of the sensor to be calibrated bymeans of a candidate calibration vector or an applied calibrationvector. The intermediate vector as a result of the encoder LSTM unitrepresents compressed information and represents the characteristics ofthe corresponding sensor behavior.

Succeeding the encoder LSTM unit, a decoder LSTM unit is applied toreceive the intermediate vector and to reconstruct the sensortrajectories from the intermediate vector such that the reconstructedtrajectories can be matched to the given reference trajectory.

The reference trajectory may be used to further compute an attention mapto determine the likelihood of what time steps of input sensortrajectories attention should be paid to. This intrinsically gives thetime delay offset estimate of the sensors.

The calibration is made for each sensor by solving an optimizationproblem to match the reconstructed sensor trajectory with the referencetrajectory by adapting the calibration vector to reduce thereconstruction error. Furthermore, during the optimization, also theencoder parameters of the encoder LSTM unit, the decoder parameters ofthe decoder LSTM unit and the attention map are optimized to obtain aparametrization of the respective sensor.

The attention map has elements that represent weights indicating therelation between each time-step of the output trajectory (output of thedecoder LSTM unit) to the time steps of the sensor trajectory (inputtrajectory) so to estimate the off-time delay using the look-backattention.

For each sensor to be calibrated, the system has to be trained by aplurality of sensor trajectories of the specific sensor and a pluralityof corresponding reference trajectories associated with the plurality ofsensor trajectories, i.e., the sensor trajectories and the referencetrajectories both are obtained during the same time frame of a system'saction. During the training the LSTM encoder/decoder neural network andparticularly the calibration vectors can be optimized.

The above described example method allows performing an automaticcalibration process for multiple sensors with respect to a referencetrajectory. By using the above configuration of the artificial neuralnetwork automatic calibration is highly robust against noise, sensorinaccuracies and off-time delays.

Furthermore, the first and second training phases may be alternatelyapplied each for a number of training cycles, particularly until aconvergence criteria is satisfied.

It may be provided that in the second training phase, an attention mapis considered which is applied on the output sensor trajectory of thedecoder function and wherein the training is configured to minimize thereconstruction error between the output sensor trajectory to which theattention map is applied and a corresponding reference trajectory.

Moreover, the encoder function and decoder function corresponds to anencoder LSTM function and a decoder LSTM function, respectively.

According to an example embodiment of the present invention, the encoderfunction may be configured to generate an intermediate vectorcorresponding to a lower dimensional manifold and the decoder functionis configured to expand the output trajectory having the same dimensionas the input trajectory a decoder LSTM function.

It may be provided that the sensor trajectory of the sensor to becalibrated and the corresponding reference trajectory are related to thesame time frame.

Furthermore, the reference trajectory may be obtained by a referencesensor wherein the reference sensor and the sensor to be calibratedbelong to the same multi-sensor system.

BRIEF DESCRIPTION OF THE DRAWINGS

Example embodiments of the present invention are described in moredetail in conjunction with the figures.

FIG. 1 shows a system with a robot with a multi-sensor arrangement todetect a trajectory of positions as an example for a system in which theautomatic calibration process can be applied on.

FIG. 2A schematically shows a conventional LSTM neural network.

FIG. 2B schematically shows a conventional LSTM neural network.

FIG. 3 schematically shows an LSTM encoder/decoder neural network usedfor automatically calibrating a sensor to a reference sensor bydetermining a calibration vector.

FIG. 4 shows a flowchart indicating the process for automaticallycalibrating the sensor of a multi-sensor system.

DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS

FIG. 1 shows exemplarily a moveable robot 1, such as a robot vacuumcleaner, which has a plurality of sensors 2 rigidly mounted with respectto the housing of the robot 1. The sensors 2 may be position sensors toprovide a sensor reading indicating position data per time instant. Forsuccessive time steps, the position data define a sensor trajectory foreach of the sensors 2. Generally, the robot 1 has a control unit 3 whichcontrols operation of the robot 1. The operation is controlled dependingon the sensor readings of the sensors 2 to perform a predefined task,for example in case of a vacuum cleaner to move the robot 1 over an areato clean the area's surface.

The sensors 2 are further referenced by indices a, b, c, d, wherein thesensor 2 a serves as a reference sensor that is the target sensor othersensors 2 b, 2 c, 2 d are calibrated towards. The reference sensor 2 aprovides sensor readings which form the reference trajectory T_(ref)within a specific time frame.

To calibrate the relative position and orientation of other sensors 2 b,2 c, 2 d which are denoted by {(r_(b),t_(b)),(r_(c),t_(c)), . . . }where r and t represent the rotation and translation vectors forcalibration of the respective sensor (defined by the index) so thatafter applying the rotation and translation vectors on the detectedsensor trajectories Tb, Tc, . . . of the corresponding sensors (obtainedduring the same time frame), they will match the reference trajectoryprovided by the reference sensor 2 a.

The sensor trajectories Tb, Tc, . . . are indicated by position data forsuccessive time steps wherein the time steps must not necessarilycorrespond to the time steps of the reference trajectory Tref providedby the reference sensor 2 a. furthermore, the sensors each have theirown sensor characteristics, provide noise and have individualinaccuracies. Therefore, it is a task to match the time step-basedtrajectories towards one another, while additionally consideringintrinsic inaccuracies and environmental noises.

According to the herein described example embodiment of the methodaccording to the present invention, this task is accomplished by meansof an encoder/decoder neural network. The encoder/decoder neural networkmay basically be formed as an LSTM encoder/decoder neural network (LSTM:long short-term memory) but can also be implemented using differentkinds of neural networks.

Substantially the ecoder/decoder neural network comprises an encoderunit followed by a decoder unit. Wherein the encoder unit compresses thetrajectory data to an intermediate vector and the decoder networkdecompresses the trajectory to reconstruct the input trajectory.Furthermore, an attention map is used to correct time related mismatchesbetween the input sensor trajectories and the corresponding referencetrajectory.

A preferred embodiment uses an LSTM neural network which is basicallydescribed in conjunction with FIGS. 2A and 2B.

FIGS. 2A and 2B schematically show an LSTM node and an illustration of arecurrent processing of timely consecutive data s_(1 . . . T) which isconventional in the art. Basically, a LSTM node 5 includes an input gate6, a forget gate 7 and an output gate 8 which are used to generate aninternal node state h_(t) and an output y_(t). By cascading multipleLSTM nodes with their internal node state h_(t) and output y_(t) torespective inputs of a succeeding LSTM node as shown in FIG. 2B, a layerof an LSTM network can be formed.

The hidden outputs h_(t=1 . . . T) from first or any of the next layerslayer can be fed to a succeeding layer as input to form a deepmulti-layer network. The number of layers generally allows to learn moreexpressive internal representation of inputs in principle.

It has been found that for the time step-based trajectory match, an LSTMencoder/decoder neural network with an attention map is well suited.Such an LSTM encoder/decoder neural network is shown in FIG. 3 and isused to calibrate one of the sensors 2 b, 2 c, 2 d to be calibrated withrespect to the reference trajectory T_(ref), e.g., provided by thereference sensor 2 a. Furthermore, the reference trajectory T_(ref) canbe provided by the control data for the motion driver of the robot orthe like. In other words if the trajectory on which the robot is movedis known from the traction control or from external observation thisdata can be used as the reference trajectory.

The configuration of FIG. 3 shows the LSTM encoder/decoder neuralnetwork 10 having an LSTM encoder unit 11 to which a calibratedtrajectory of a sensor to be calibrated is fed. The an LSTM encoder unit11 has a number of an LSTM encoder cells 12 each of which is formed as aLSTM cell as described above. The LSTM encoder cells 12 are sequentiallycoupled and arranged in a number of consecutive layers.

The calibrated trajectory is formed by a number of timely succeedingsensor readings s_(1 . . . T) and obtained by applying a candidatecalibration vector (r_(cand),t_(cand)) (for training) or a previouslydetermined calibration vector (r,t) (for a trained system) onto eachsensor reading value s_(1 . . . T) of the different time instances t=1 .. . T in a respective calibration unit 13. The data items of thecalibrated sensor reading values s′_(1 . . . T) at successive time stepsare fed to a first layer formed by a number T of LSTM cells 12 asdescribed above.

The trajectory data is propagated through a number of layers of LSTMcells 12 while the output data of the last layer of LSTM cells of theencoder LSTM unit 11 is compressed to a lower dimensional manifold, suchas an intermediate vector z.

The intermediate vector z is generated by the last LSTM cell 12 of thelast layer of the encoder LSTM unit 11. The last LSTM cell 12 hasimplemented a function mapping input which maps to a lower dimensionintermediate vector z by multiplying a matrix with size n*m, where thedimension n is the size of h_(T)/y_(T) and where the dimension m is muchsmaller than n (original input dimension).

The output y_(T) of the last LSTM cell 12 is transformed by a givennon-linear transformation to obtain the intermediate vector z.

The intermediate vector z is serially transferred to the input of afirst LSTM cell 12 of the decoder unit. For instance, the non-lineartransformation may be

z=σ(wy ^(T) +b).

which uses the output y^(T) of last LSTM cell where σ(⋅) is anactivation function, e.g., sigmoid, which is nonlinear.

The projection mentioned above is all computed automatically by errorbackpropagation, since the weights of the LSTM cells 12 are optimized bycomputing the z vector to recover trajectories as being base frames.

The intermediate vector z includes the data of the calibrated trajectoryin a compressed manner, so that the data of the intermediate vector zrepresent the sensor trajectory.

The intermediate vector z is thereafter supplied to a decoder LSTM unit14 which has a plurality of layers of a number T of LSTM cells 15sequentially arranged. The decoder LSTM unit 14 has the same time stepdimension T as the encoder LSTM unit 11 so that the calibratedtrajectory at the input of the encoder LSTM unit 11 can be reconstructedfrom the intermediate vector z.

In an attention map unit 16 the reconstructed trajectory is applied toan attention map to compensate for a time mismatch between the time baseof the reference trajectory T_(ref) and the trajectory T_(b), T_(c),T_(d) to be calibrated. Informally, an attention map equips a neuralnetwork with the ability to focus on a subset of its inputs, i.e., on asubset of time steps.

Comparing the resulting reconstructed trajectory with the referencetrajectory leads to an error which can be used to train the calibrationvector, the encoder and decoder function and the attention map. Theoptimization problem can be written as:

η(r,t),f ^(decoder) ,g _(encoder) ,W˜min Σ_(t) ∥s _(a) ^(t) −f^(decoder)(g _(encoder)(η_((r,t))(s _(k˜(b,c, . . . m)) ^(t))))+W _(t)^(T) s _(k˜(b,c, . . . m))∥²

wherein s^(t) _(a) indicates the sensor reading values of a time step tfor the reference sensor 2 a, η(r,t) the calibration vector, W^(T) _(t)an attention map and g_(encoder) the encoder function and f^(decoder)the decoder function.

The encoder LSTM unit 11 and the decoder LSTM unit 14 can be trained ina training phase by feeding data batches including sensor trajectoriesand reference trajectories and by training above parameters.Particularly, a first training phase includes the application of thereference trajectory without using the calibration units 13 (or byswitching them to not rendering the respective reference trajectorydata) to train the encoder and decoder function by minimizing thereconstruction error at the output of the decoder LSTM unit 14. Theattention map unit 16 may be either deactivated or activated and appliedwith no time offset with respect to the reference trajectory.

In the second training phase as the attention map is input dependent,for each pair of a sensor trajectory and corresponding referencetrajectory. However, as in the first training phase only the referencetrajectory is applied which has no time offset with respect to itselfthe attention map can be deactivated.

In a second training phase a sensor trajectory to be calibrated isapplied on the inputs of the calibration units and the encoder LST unitand the decoder LSTM unit 14 as well as the attention map are trained toreduce the reconstruction error. The reconstruction error is determinedby a distance between the reconstructed trajectory (which is obtained byusing calibration units, the encoder LSTM unit, decoder LSTM unit andthe attention map unit) and the corresponding reference trajectory.

Both training phases can be alternately carried out each for a pluralityof training cycles.

In the flowchart of FIG. 4, the process of calibration of the sensors 2of the sensor system of the aforementioned robot 1 is described.

In step S1, among the sensors 2 of the sensor system, a reference sensor2 a is selected. All other sensors 2 b, 2 c, 2 d will be calibrated withrespect to the selected reference sensor 2 a.

In step S2, trajectories are recorded for each of the sensors 2, e.g.,by moving the robot along a random or predetermined path. Thereby, areference trajectory Tref can be obtained from the sensor readings ofthe reference sensor 2 a and further sensor trajectories can be obtainedfrom the other sensors 2 b, 2 c, 2 d. Alternatively, sensor trajectoriescan be obtained from moving the robot 1 along one or more known motionpaths which correspond to a respective reference trajectory. The numberof training trajectories is selected so that the set of trainingtrajectory data is sufficient to train the LSTM encoder/decoder neuralnetwork 10 as described above.

In step S3, the first training phase is performed. Therein, the variousreference trajectories are used to train the LSTM encoder/decodernetwork 10 by minimizing the reconstruction error of propagating each ofthe reference trajectories through the encoder LSTM unit 11 and thedecoder LSTM unit 14 according to above formula. The reconstructionerror can be e.g. defined as a distance between the reconstructedtrajectory at an output trajectory of the decoder LSTM unit 14 and thereference trajectory. Thereby, parameters for the encoder functionf_(encoder) and the decoder function g^(decoder) can be trained applyinga conventional method such as by backpropagation.

Following the training related to the reference trajectories, one of thesensors 2 b, 2 c, 2 d to be calibrated is selected in step S4 andinitially a candidate calibration vector (r,t) (as an initialcalibration vector) is assumed. By means of the calibration unit thecandidate calibration vector is applied on the sensor reading values ofthe sensor trajectory of the sensor 2 b, 2 c, 2 d to be calibratedbefore supplying them to the input of the encoder LSTM unit 11.

The second training phase is carried out by propagating the calibratedsensor trajectories through the LSTM encoder/decoder neural network 10and a reconstructed trajectory is respectively obtained at the output ofthe attention map unit. The reconstruction error between thereconstructed trajectory and the reference trajectory belonging to thesame time frame is used to optimize the calibration vector, the encoderfunction, the decoder function and the attention matrix W for therespective sensor 2 b, 2 c, 2 d.

In step S5 it is checked it the training process for the various sensortrajectories of the respective sensor has been completed. This can bedone by applying a respective convergence criteria, by reaching a numberof iterations or the like. If the training process has not beencompleted (alternative: yes) it is returned to step S3. Otherwise(alternative: no) the process is continued with step S6. The above stepsS3, S4 of optimization are repeated for the various motion paths (timeframes), i.e. the various sensor trajectories for which sensortrajectories of the respective (selected) sensor and associatedreference trajectories are available.

In step S6 it is checked if a next sensor 2 of the sensor system shallbe calibrated. If there is a next sensor to be calibrated (alternative:yes) then the method is continued by step S3 using the various sensortrajectories T of the corresponding next sensor. Otherwise (alternative:no), the process is discontinued.

The so obtained calibration vector W^(T) _(t) can be applied on thesensor reading of the respective sensor associated with the respectivecalibration vector W^(T) _(t) by the product of W^(T) _(t) s^(t)_(b,c,d) for sensors 2 b, 2 c, 2 d.

What is claimed is:
 1. A method for calibrating a sensor with respect toa reference trajectory indicating a number of timely successive datavalues using an encoder/decoder neural network with an encoder functionand a successive decoder function, the method comprising the steps of:training, in a first training phase, the encoder function and thedecoder function of the encoder/decoder neural network with respect to areference trajectory so that a reconstruction error of the referencetrajectory at an input of the encoder function and a reconstructedreference trajectory at an output of the decoder function is minimized;training, in a second training phase, a calibration vector with respectto a sensor trajectory so that a reconstruction error between an outputsensor trajectory and a corresponding reference trajectory is minimized,wherein the output sensor trajectory is obtained by applying thecalibration vector on the sensor trajectory and by applying the encoderfunction and decoder function on the sensor trajectory; and applying thecalibration vector to calibrate the sensor.
 2. The method according toclaim 1, wherein the first training phase and the second training phaseare alternately applied each for a number of training cycles until aconvergence criteria is satisfied.
 3. The method according to claim 1,wherein the step of training in the second training phase includestraining of the calibration vector, the encoder function, and thedecoder function, of the encoder/decoder neural network.
 4. The methodaccording to claim 1, wherein in the second training phase, an attentionmap is considered which is applied on the output sensor trajectory ofthe decoder function, and wherein the training in the second trainingphase is configured to minimize the reconstruction error between theoutput sensor trajectory to which the attention map is applied and acorresponding reference trajectory.
 5. The method according to claim 1,wherein the encoder function and decoder function corresponds to anencoder LSTM function and a decoder LSTM function, respectively.
 6. Themethod according to claim 1, wherein the encoder function is configuredto generate an intermediate vector corresponding to a lower dimensionalmanifold, and the decoder function is configured to expand the outputtrajectory having a same dimension as the input trajectory.
 7. Themethod according to claim 1, wherein the sensor trajectory of the sensorto be calibrated and the corresponding reference trajectory are relatedto the same time frame.
 8. The method according to claim 1, wherein thereference trajectory is obtained by a reference sensor wherein thereference sensor and the sensor to be calibrated belong to the samemulti-sensor system.
 9. A sensor system, comprising: multiple sensorsfor obtaining sensor trajectories; an encoder/decoder neural networkwith an encoder function and a successive decoder function; a controlunit configured to calibrate at least one of the multiple sensors withrespect to a reference trajectory by: training, in a first trainingphase, the encoder function and the decoder function of theencoder/decoder neural network with respect to a reference trajectory sothat a reconstruction error of the reference trajectory at an input ofthe encoder function and a reconstructed reference trajectory at anoutput of the decoder function is minimized; training, in a secondtraining phase, a calibration vector with respect to a sensor trajectoryso that a reconstruction error between an output sensor trajectory and acorresponding reference trajectory is minimized, wherein the outputsensor trajectory is obtained by applying the calibration vector on thesensor trajectory and by applying the encoder function and decoderfunction on the sensor trajectory; and applying the calibration vectorto calibrate the sensor.
 10. A non-transitory machine readable medium onwhich is recorded a computer program for calibrating a sensor withrespect to a reference trajectory indicating a number of timelysuccessive data values using an encoder/decoder neural network with anencoder function and a successive decoder function, the computerprogram, when executed by a computer, causing the computer to perform:training, in a first training phase, the encoder function and thedecoder function of the encoder/decoder neural network with respect to areference trajectory so that a reconstruction error of the referencetrajectory at an input of the encoder function and a reconstructedreference trajectory at an output of the decoder function is minimized;training, in a second training phase, a calibration vector with respectto a sensor trajectory so that a reconstruction error between an outputsensor trajectory and a corresponding reference trajectory is minimized,wherein the output sensor trajectory is obtained by applying thecalibration vector on the sensor trajectory and by applying the encoderfunction and decoder function on the sensor trajectory; and applying thecalibration vector to calibrate the sensor.